To find the best-fit circle (also known as the least-squares circle) through a set of weighted points $(x_i, y_i, w_i)$, we need to determine the circle's center $(h, k)$ and radius $r$. The equation of the circle is:

$(x - h)^2 + (y - k)^2 = r^2$

Given points:

$(2,0,1), \quad (0,1,1), \quad (-1,0,1), \quad (0,-1,1)$

Steps to find the best-fit circle:

1. Set up the system using the general circle equation: $x^2 + y^2 + Dx + Ey + F = 0$ where $D = -2h$, $E = -2k$, and $F = h^2 + k^2 - r^2$.

2. Construct the linear system: Substitute each point $(x_i, y_i)$ into the equation:

   $x_i^2 + y_i^2 + D x_i + E y_i + F = 0$

   This gives four linear equations.

3. Solve for $D, E, F$ using least-squares minimization (since we have more equations than unknowns).

4. Extract the center $(h, k)$ and radius $r$: $h = -D/2, \quad k = -E/2, \quad r = \sqrt{h^2 + k^2 - F}$

Let me compute the exact values for you. numpy as np

Given points (x, y, w) where w is the weight

points = np.array([ [2, 0, 1], [0, 1, 1], [-1, 0, 1], [0, -1, 1] ])

Extract x, y coordinates (ignoring weights for now)

x = points[:, 0] y = points[:, 1]

Constructing matrix A and vector b for least squares fitting

A = np.column_stack((x, y, np.ones_like(x))) b = -(x2 + y2)

Solve for D, E, F using least squares

params, _, _, _ = np.linalg.lstsq(A, b, rcond=None) D, E, F = params

Compute the center (h, k) and radius r

h = -D / 2 k = -E / 2 r = np.sqrt(h2 + k2 - F)

(h, k, r)The best-fit circle has:

• Center: $(0.553, 0)$
• Radius: $1.334$

So, the equation of the best-fit circle is:

$(x - 0.553)^2 + y^2 = (1.334)^2$

Would you like a visualization of the circle with the points plotted?

Here are five related questions to expand on this topic:

1. How do you find a best-fit circle using least squares for more complex datasets?
2. How does weighting affect the least-squares circle fitting method?
3. What are the practical applications of best-fit circles in real-world problems?
4. How does this method compare to geometric circle fitting (like Kasa’s method)?
5. How can this be extended to best-fit spheres in 3D?

Tip:

When fitting circles using least squares, using more points improves accuracy, but outliers can strongly influence the result. Consider using robust regression techniques to minimize their impact.